文档介绍：A2Preview2of2Calculus
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A¡ The origins of calculus go back at least 2500 years to the ancient Greeks, who found
areas using the “method of exhaustion.” They knew how to find the area A of any poly-
A∞
A™ gon by dividing it into triangles as in Figure 1 and adding the areas of these triangles.
A£ A¢ It is a much more difficult problem to find the area of a curved figure. The Greek
method of exhaustion was to inscribe polygons in the figure and circumscribe poly-
gons about the figure and then let the number of sides of the polygons increase.
A=A¡+A™+A£+A¢+A∞
Figure 2 illustrates this process for the special case of a circle with inscribed regular
FIGURE 1 polygons.
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A¡™ии
FIGURE 2
Let An be the area of the inscribed polygon with n sides. As n increases, it appears
that An es closer and closer to the area of the circle. We say that the area of the
The4Preview4Module4is4a4numeri- circle is the limit of the areas of the inscribed polygons, and we write
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A
lim An
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Greeks themselves did not use limits explicitly. However, by indirect reasoning,
Eudoxus (fifth century .) used exhaustion to prove the familiar formula for the area
of a circle: A
r 2.
We will use a similar idea in Chapter 5 to find areas of regions of the type shown
in Figure 3. We will approximate the desired area A by ar